Notes
Topic 3-6: Chain Rules for Multivariate
Functions
Textbook: Section 14.5, 14.6
Big Ideas
I
Notes
The chain rule is used to compute the derivatives of
compositions of functions.
A New Notation for Curves
I
Suppose a curve is parameterized by the vector-valued
function
~
r (t) = x(t), y (t), z(t) .
I
The points on the curve (the terminal points of ~
r (t)) will
be denoted:
c(t) = x(t), y (t), z(t) .
I
The derivative of c(t) is defined to be the tangent vector.
So, in the textbook and WebAssign:
I
When you see “c(t)”, think “~
r (t)”.
I
When you see “~c 0 (t)00 , think “~
r 0 (t)” (or “~
v (t)”, for
velocity).
Notes
Recall: The Chain Rule from Calc. 1
Notes
Chain rules are used to taking derivatives of compositions of
functions.
I
Recall from Calc I:
The derivative of the composition f u(t) is:
df du
d
f u(t) =
dt
du dt
In words:
I
Multiply the derivative of the “outside function” f taken
with respect to the “inside function” u, by the derivative
of the “inside function” u taken with respect to t.
The Chain Rule Along Paths (One Parameter)
Notes
If ~
r (t) = x(t), y (t), z(t) is a differentiable curve in the
domain of a differentiable function f (x, y , z), then:
The derivative of f (x, y , z) restricted to the curve ~
r (t) is:
d r
~ · d~
f ~
r (t) = ∇f
dt
dt
~ ·~
= ∇f
r 0 (t)
=
∂f dx
∂f dy
∂f dz
+
+
∂x dt
∂y dt
∂z dt
Example: Chain Rule Along Paths (One Variable)
function: f (x, y ) = x 2 y 3
path: ~
r (t) = t, t 2
d f ~
r (t) using the chain rule for paths.
dt
I
Compute
I
What is the rate of change of f along the path at t = 2?
Notes
Example: Chain Rule Along Paths; the Path is Not
Given Explicitly
I
Notes
An ant is skating on the xy -plane. You don’t know her
path, but you do know that her velocity as she passes
through the point P = (−1, 2) is ~
v = 3, −5 .
Suppose T (x, y ) = x 2 y 3 represents the temperature at a
point P = (x, y ). What is the rate of change of
temperature with respect to time, dT /dt, experienced by
the ant as she passes through the point P?
What Does the Chain Rule Mean?
Notes
Returning to the previous example: To compute dT /dt, we
took the dot product of the gradient of the function with the
~ ·~
ant’s velocity: dT /dt = ∇T
v.
I
If we rewrite velocity as speed times the unit vector in the
direction of the velocity, ~
v = k~
v kv̂ , we see:
dT
~ · k~
= ∇T
v kv̂
dt
~ · v̂
= k~
v k ∇T
= k~
v kDv̂ T
So the chain rule along paths can be interpreted as the
directional derivative in the direction of the velocity
(tangent vector to the path), multiplied by the speed of
travel along the path!
Chain Rule in Several Variables
If f (x, y ), x(s, t), y (s, t), and z(s, t) are all differentiable,
with x(s, t), y (s, t), and z(s, t) in the domain of f , then:
∂f
r
∂f ∂x
∂f ∂y
∂f ∂z
~ · ∂~
= ∇f
=
+
+
∂s
∂s
∂x ∂s
∂y ∂s
∂z ∂s
∂f
r
∂f ∂x
∂f ∂y
∂f ∂z
~ · ∂~
= ∇f
=
+
+
∂t
∂t
∂x ∂t
∂y ∂t
∂z ∂t
Notes
Example: Computing the Chain Rule in Several
Variables
Notes
f (x, y ) = ye x
x(s, t) = st,
y (s, t) = s + t 2
∂f
∂f
and
.
∂s
∂t
I
Find
I
Evaluate
∂f
at the point (s, t) = (0, 1).
∂s
When Would You Use the Chain Rule in Several
Variables?
Notes
Suppose the temperature at a point on the xy -plane is given
by a function T (x, y ), which depends on your location in
Cartesian cooridinates.
But what if you want to work in polar coordinates? You can
re-parameterize the plane using the coordinate functions:
x(r , θ) = r cos θ,
I
y (r , θ) = r sin θ.
Then, for example the rate of change of temperature with
respect to a change in the angle θ is:
∂T
~ (x, y ) · ∂x , ∂y
= ∇T
∂θ
∂θ ∂θ
Worth Noting
Notes
All derivatives of a function of more than one variable can be
computed by taking the dot product of the gradient with the
appropriate vector:
I Directional derivative:
~ · û
Dû f = ∇f
I
Chain rule in a singe variable (along the curve ~
r (t)):
df
~ ·~
= ∇f
r 0 (t)
dt
I
Chain rule in more than one variable:
∂f
~ ·~
= ∇f
r s (s, t)
∂s